- Kuratowski closure axioms
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In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces.
A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.
Contents
Definition
A topological space is a set X with a function
called the closure operator where is the power set of X.
The closure operator has to satisfy the following properties for all
- (Extensivity)
- (Idempotence)
- (Preservation of binary unions)
- (Preservation of nullary unions)
If the second axiom, that of idempotence, is relaxed, then the axioms define a preclosure operator.
Notes
By induction, Axioms (3) and (4) are equivalent to the single statement
- (Preservation of finitary unions).
Recovering topological definitions
A function between two topological spaces
is called continuous if for all subsets A of X
A point p is called close to A in if
A is called closed in if . In other words the closed sets of X are the fixed points of the closure operator.
If one takes an "open set" to be a set whose complement is closed, then the family of all open sets forms a topology. Conversely, any topology can be induced in this way by the correct choice of closure operator.
See also
Categories:- Closure operators
- Mathematical axioms
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